Approximate Graph Spectral Decomposition with the Variational Quantum Eigensolver

TL;DR

This paper extends the Variational Quantum Eigensolver to analyze spectral properties of graphs, demonstrating its effectiveness on quantum hardware and showing potential superpolynomial runtime advantages over classical methods.

Contribution

It adapts VQE for spectral graph analysis, evaluates its performance on quantum hardware, and compares it to classical algorithms, highlighting potential quantum speedups.

Findings

Successfully analyzed graphs with up to 64 vertices on quantum hardware.

Demonstrated superpolynomial runtime improvement over classical algorithms.

Provided empirical and theoretical comparisons of different ansatz and graph parameters.

Abstract

Spectral graph theory is a branch of mathematics that studies the relationships between the eigenvectors and eigenvalues of Laplacian and adjacency matrices and their associated graphs. The Variational Quantum Eigensolver (VQE) algorithm was proposed as a hybrid quantum/classical algorithm that is used to quickly determine the ground state of a Hamiltonian, and more generally, the lowest eigenvalue of a matrix $M\in \mathbb{R}^{n\times n}$. There are many interesting problems associated with the spectral decompositions of associated matrices, such as partitioning, embedding, and the determination of other properties. In this paper, we will expand upon the VQE algorithm to analyze the spectra of directed and undirected graphs. We evaluate runtime and accuracy comparisons (empirically and theoretically) between different choices of ansatz parameters, graph sizes, graph densities, and matrix types, and demonstrate the effectiveness of our approach on Rigetti's QCS platform on graphs of up to 64 vertices, finding eigenvalues of adjacency and Laplacian matrices. We finally make direct comparisons to classical performance with the Quantum Virtual Machine (QVM) in the appendix, observing a superpolynomial runtime improvement of our algorithm when run using a quantum computer.